Problem Set N°1
Valentina Andrade
August 27, 2011
Abstract
The following report contains the exercises requested in problem set 1. In the first part you can download the proofs of some properties and/or results related to AR, MA and ARMA process. In the second part, the Box-Jenkins methodology is applied to study three series of the Chilean economy: inflation, exchange rate and IPSA. One of the most important results of both exercises is related to how to apprehend time series structures, either theoretically or empirically we can say something that Wold ‘s theorem had already anticipated’‘Any stationary series can beexpressed as the sum of two components: a perfectly forecastable series and a moving average of possily infinite order’’Part 2
ARMA models have been presented as a parsimonious tool to describe non-stationary stochastic processes. In theory, non-stationary series can be represented by an MA(\(\infty\)), i.e., capturing the entire memory of the series.
In practice this is very expensive, so we will show how we can approximate an MA(\(\infty\)) from an ARMA(\(p,q\)) model, with few parameters (i.e. \(p+q\) is small). We will be guided by the methodology of Box and Jenkins to achieve this task.
In order to use ARMA we need the non-stationary components or “trends around the mean” or “trends around the variance” to be removed. In addition to using transformations, we test a unit root test (Dickey Fuller’s test).
Other deterministic components are removed. In our case this is important because before 2001 we find that there is a clear inflationary path, and that this is evidently due to the change in the monetary policy regime (3% rule).
Third, we compute ACF and PACF to identify the order and type of the underlying model.
The model is estimated assuming the proposed model with p and q.
Identification tests are performed and the adequacy of the identification is evaluated. In this report we give importance to AIC and Ljung-Box.
In-sample predictions of the estimated model are made.
Data exploration
Figure 1. Series infsv_sa (IPC),
IPSA_sa (IPSA), tcn_sa (Exchange Rate CLP/USD)
(1990- 2022)
As can be shown in Figure 1, inflation (measured by the consumer price index) presents a clear trend before 2001. This price growth trend was stabilized after the Central Bank set a target of around 3% inflation and a policy of nominalization (Fuentes et al, 2003). Similarly, since 2020, due to the health crisis caused by the COVID-19 pandemic, the consequences have also been reflected in an increase in the cost of living.
In order to isolate the trends mentioned above, we have chosen to limit the period of analysis from 2001 to 2020, both for inflation and for the other variables of interest, in order to make the models more comparable. We will use the series shown in Figure 2 for the following steps.
Figure 2. Series infsv_sa (Inflation),
ipsa_sa (IPSA), tcn_sa (Exchange
Rate CLP/USD) (2001- 2022)
1. Inflation
| method | Valor-p | statistic | parameter | alternative | resultado 95% |
|---|---|---|---|---|---|
| Augmented Dickey-Fuller Test | 0.2727477 | -2.720626 | 6 | stationary | Existe unit-root |
| method | Valor-p | statistic | parameter | alternative | resultado 95% |
|---|---|---|---|---|---|
| Augmented Dickey-Fuller Test | 0.01 | -7.911852 | 6 | stationary | Es I(0), no unit-root |
| Ajuste | sigma | logLik | AIC | BIC | Box-Ljung test residuos p value |
|---|---|---|---|---|---|
| ARMA(6, 6) | 0.1242021 | 161.6445 | -295.2890 | -246.5600 | 0.9528551 |
| ARMA(3, 4) | 0.1266875 | 156.4905 | -294.9810 | -263.6553 | 0.9888481 |
| ARMA(5, 2) | 0.1278088 | 156.1140 | -294.2279 | -262.9022 | 0.9870518 |
| ARMA(6, 5) | 0.1265816 | 159.7020 | -293.4041 | -248.1558 | 0.7037629 |
| ARMA(5, 3) | 0.1280881 | 156.1609 | -292.3218 | -257.5155 | 0.9719941 |
| ARMA(4, 3) | 0.1287374 | 154.7641 | -291.5282 | -260.2024 | 0.4883086 |
| ARMA(5, 5) | 0.1260530 | 157.6999 | -291.3998 | -249.6321 | 0.8613235 |
| ARMA(6, 3) | 0.1280977 | 156.5268 | -291.0536 | -252.7666 | 0.9612549 |
| ARMA(5, 4) | 0.1283396 | 156.1747 | -290.3494 | -252.0624 | 0.9737545 |
| ARMA(4, 6) | 0.1272511 | 157.1120 | -290.2240 | -248.4563 | 0.7843708 |
| ARMA(6, 4) | 0.1283190 | 156.6965 | -289.3930 | -247.6254 | 0.7887453 |
| ARMA(1, 1) | 0.1310111 | 148.5588 | -289.1176 | -275.1951 | 0.9331472 |
We select the model Modelo ARMA(6, 6), el cual posee AIC de -295.2889897. The estimated parameters are:
| term | estimate | std.error | 2.5 % | 97.5 % |
|---|---|---|---|---|
| ar1 | 0.6141906 | 0.1926052 | 0.2366914 | 0.9916899 |
| ar2 | -0.4191249 | 0.1031636 | -0.6213219 | -0.2169279 |
| ar3 | -0.3976778 | 0.1096910 | -0.6126683 | -0.1826873 |
| ar4 | 0.5634687 | 0.0950703 | 0.3771343 | 0.7498030 |
| ar5 | -0.9534417 | 0.0919981 | -1.1337546 | -0.7731288 |
| ar6 | 0.1074972 | 0.1648675 | -0.2156372 | 0.4306317 |
| ma1 | -1.1873407 | 0.1901318 | -1.5599922 | -0.8146892 |
| ma2 | 0.7616632 | 0.1995061 | 0.3706384 | 1.1526880 |
| ma3 | 0.1725191 | 0.2204035 | -0.2594639 | 0.6045021 |
| ma4 | -0.8491095 | 0.1941380 | -1.2296130 | -0.4686060 |
| ma5 | 1.3350735 | 0.1818911 | 0.9785734 | 1.6915736 |
| ma6 | -0.5293383 | 0.1686824 | -0.8599498 | -0.1987268 |
| intercept | -0.0009994 | 0.0037103 | -0.0082714 | 0.0062727 |
Auto-correlation functions are represented in ACF and PACF, to figure the order of MA and AR models, respectively.
IPSA
| method | Valor-p | statistic | parameter | alternative | resultado 95% |
|---|---|---|---|---|---|
| Augmented Dickey-Fuller Test | 0.01 | -5.043973 | 6 | stationary | Es I(0), no unit-root |
| Ajuste | sigma | logLik | AIC | BIC | Box-Ljung test residuos p value |
|---|---|---|---|---|---|
| ARMA(3, 3) | 4.065532 | -678.7555 | 1373.511 | 1401.389 | 0.7484041 |
| ARMA(1, 1) | 4.164760 | -684.2937 | 1376.587 | 1390.527 | 0.4704815 |
| ARMA(4, 2) | 4.119725 | -680.3639 | 1376.728 | 1404.606 | 0.9895842 |
| ARMA(1, 2) | 4.163729 | -683.7363 | 1377.473 | 1394.897 | 0.9721260 |
| ARMA(2, 1) | 4.164388 | -683.7731 | 1377.546 | 1394.970 | 0.9903279 |
| ARMA(5, 5) | 4.062058 | -676.9303 | 1377.861 | 1419.678 | 0.9573418 |
| ARMA(6, 2) | 4.092606 | -679.1165 | 1378.233 | 1413.081 | 0.9968753 |
| ARMA(4, 3) | 4.127846 | -680.2603 | 1378.521 | 1409.884 | 0.9887073 |
| ARMA(5, 2) | 4.129013 | -680.3567 | 1378.713 | 1410.077 | 0.9896419 |
| ARMA(1, 3) | 4.170762 | -683.6374 | 1379.275 | 1400.184 | 0.9715892 |
| ARMA(2, 2) | 4.172423 | -683.7272 | 1379.454 | 1400.363 | 0.8881015 |
| ARMA(3, 1) | 4.172472 | -683.7327 | 1379.465 | 1400.374 | 0.9854579 |
We select the model Modelo ARMA(6, 6), el cual posee AIC de 1381.8242813. The estimated parameters are:
| term | estimate | std.error | 2.5 % | 97.5 % |
|---|---|---|---|---|
| ar1 | 0.3453801 | 0.7869316 | -1.1969775 | 1.8877377 |
| ar2 | 0.0834383 | 0.1748760 | -0.2593123 | 0.4261890 |
| ar3 | -0.2847331 | 0.1346767 | -0.5486945 | -0.0207717 |
| ar4 | -0.4079086 | 0.2029009 | -0.8055870 | -0.0102303 |
| ar5 | 0.7666452 | 0.3684773 | 0.0444430 | 1.4888474 |
| ar6 | -0.1350985 | 0.5635310 | -1.2395990 | 0.9694019 |
| ma1 | -0.3225307 | 0.8049784 | -1.9002593 | 1.2551979 |
| ma2 | 0.0050931 | 0.1894970 | -0.3663142 | 0.3765004 |
| ma3 | 0.3259910 | 0.1247129 | 0.0815583 | 0.5704237 |
| ma4 | 0.5405318 | 0.2568678 | 0.0370803 | 1.0439834 |
| ma5 | -0.8062602 | 0.4755525 | -1.7383260 | 0.1258055 |
| ma6 | 0.1538766 | 0.5949032 | -1.0121122 | 1.3198654 |
| intercept | 0.6322126 | 0.3610376 | -0.0754082 | 1.3398333 |
Auto-correlation functions of residuals are represented in ACF and PACF, to figure the order of MA and AR models, respectively.
3. Exchange Rate
| method | Valor-p | statistic | parameter | alternative | resultado 95% |
|---|---|---|---|---|---|
| Augmented Dickey-Fuller Test | 0.01 | -5.594978 | 6 | stationary | Es I(0), no unit-root |
| Ajuste | sigma | logLik | AIC | BIC | Box-Ljung test residuos p value |
|---|---|---|---|---|---|
| ARMA(3, 2) | 2.380310 | -548.1035 | 1110.207 | 1134.600 | 0.9592174 |
| ARMA(4, 2) | 2.375450 | -547.1140 | 1110.228 | 1138.106 | 0.9793663 |
| ARMA(2, 3) | 2.381598 | -548.2437 | 1110.487 | 1134.881 | 0.9573739 |
| ARMA(3, 3) | 2.377868 | -547.3355 | 1110.671 | 1138.549 | 0.8419124 |
| ARMA(2, 5) | 2.375085 | -546.5865 | 1111.173 | 1142.536 | 0.9521570 |
| ARMA(1, 1) | 2.402211 | -551.7083 | 1111.417 | 1125.356 | 0.9867599 |
| ARMA(5, 2) | 2.378433 | -546.8993 | 1111.799 | 1143.162 | 0.9789216 |
| ARMA(2, 2) | 2.393469 | -549.9395 | 1111.879 | 1132.788 | 0.2429308 |
| ARMA(4, 3) | 2.380006 | -547.0524 | 1112.105 | 1143.468 | 0.9945415 |
| ARMA(2, 4) | 2.385934 | -548.1558 | 1112.312 | 1140.190 | 0.9863368 |
| ARMA(1, 2) | 2.403171 | -551.3031 | 1112.606 | 1130.030 | 0.9994660 |
| ARMA(2, 6) | 2.378219 | -546.3872 | 1112.774 | 1147.622 | 0.9952449 |
We select the model Modelo ARMA(3, 2), el cual posee AIC de 1110.2068976. The estimated parameters are:
| term | estimate | std.error | 2.5 % | 97.5 % |
|---|---|---|---|---|
| ar1 | 0.4944275 | 0.1949472 | 0.1123380 | 0.8765171 |
| ar2 | -0.8592603 | 0.1104293 | -1.0756978 | -0.6428228 |
| ar3 | 0.1498004 | 0.0822274 | -0.0113624 | 0.3109632 |
| ma1 | -0.2140941 | 0.1762483 | -0.5595344 | 0.1313461 |
| ma2 | 0.8647872 | 0.0940674 | 0.6804185 | 1.0491559 |
| intercept | 0.1829407 | 0.2054703 | -0.2197736 | 0.5856550 |
Auto-correlation functions of residuals are represented in ACF and PACF, to figure the order of MA and AR models, respectively.
